Preheat oven to 375°F (190°C).

In a large skillet, sauté the chicken, carrots, potatoes, onion, and garlic until the chicken is cooked through and the vegetables are tender.

Add the flour, chicken broth, milk, thyme, salt, and pepper to the skillet. Stir until the mixture thickens.

Transfer the mixture to the baking dish.

To make the crust, combine the flour and salt in a bowl. Add the butter and use a pastry blender or your fingers to work it into the flour until the mixture resembles coarse crumbs.

Gradually add the ice water, stirring with a fork until the dough comes together.

Roll out the dough on a floured surface to a size slightly larger than the baking dish.

Place the dough over the filling and trim the edges. Crimp to seal.

Cut a few slits in the top crust to allow steam to escape.

Brush the top crust with the beaten egg and sprinkle with water.

Bake for 35 minutes, or until the crust is golden brown.